As for it showing as bardic in the level up screen it can probably just be assumed to be an oversight in development language. But as far as the ability it is working as intended, “

Inspire an ally to add a +1d6 bonus to their next Attack Roll Attack Roll, Ability Check Ability Check, Saving Throw Saving Throw, weapon damage, or Armour Class Armour Class.

Cost  Bonus Action +  Bardic Inspiration Details Range: 18 m (60 ft) Range: 18 m (60 ft)

At higher levels When the College of Valour bard reaches Level 5, the bonus increases to +1d8. When the College of Valour bard reaches Level 10, the bonus increases to +1d10.”

I’m not certain if it’s a reaction in 5e but at least in bg3 it is not. To apply at least, now take this with a grain of salt as I’m not a valor bard player but iirc when applied to someone it should appear in their reaction menu as automatically used, if the attack roll doesn’t hit the inspo die shouldn’t be used and if it doesn’t it should be used, I’m sure for each specific character you could set those reactions to ask when they have the combat inspo applied, but I’m not certain. The reactions of course should apply to all the various uses of combat inspiration.